Question: Solve for $z$, $ \dfrac{5z - 5}{8z + 12} = \dfrac{10}{2z + 3} + \dfrac{8}{6z + 9} $
Solution: First we need to find a common denominator for all the expressions. This means finding the least common multiple of $8z + 12$ $2z + 3$ and $6z + 9$ The common denominator is $24z + 36$ To get $24z + 36$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ \dfrac{5z - 5}{8z + 12} \times \dfrac{3}{3} = \dfrac{15z - 15}{24z + 36} $ To get $24z + 36$ in the denominator of the second term, multiply it by $\frac{12}{12}$ $ \dfrac{10}{2z + 3} \times \dfrac{12}{12} = \dfrac{120}{24z + 36} $ To get $24z + 36$ in the denominator of the third term, multiply it by $\frac{4}{4}$ $ \dfrac{8}{6z + 9} \times \dfrac{4}{4} = \dfrac{32}{24z + 36} $ This give us: $ \dfrac{15z - 15}{24z + 36} = \dfrac{120}{24z + 36} + \dfrac{32}{24z + 36} $ If we multiply both sides of the equation by $24z + 36$ , we get: $ 15z - 15 = 120 + 32$ $ 15z - 15 = 152$ $ 15z = 167 $ $ z = \dfrac{167}{15}$